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G = C2×C4.10C42  order 128 = 27

Direct product of C2 and C4.10C42

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4.10C42, C24.6Q8, (C2×C4).88C42, C4.41(C2×C42), (C22×C8).17C4, C23.49(C2×Q8), (C22×C4).34Q8, C23.58(C4⋊C4), (C22×C4).253D4, C4○(C4.10C42), (C2×M4(2)).24C4, (C23×C4).216C22, (C22×C4).645C23, (C22×M4(2)).9C2, C4.14(C2.C42), (C2×M4(2)).294C22, C22.27(C2.C42), (C2×C8).8(C2×C4), C22.7(C2×C4⋊C4), (C2×C4).225(C2×D4), C4.79(C2×C22⋊C4), (C2×C4).120(C4⋊C4), (C2×C4).515(C22×C4), (C22×C4).478(C2×C4), C2.8(C2×C2.C42), (C2×C4).112(C22⋊C4), SmallGroup(128,463)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×C4.10C42
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×C4.10C42
Lower central
C1 — C4 — C2×C4.10C42
Upper central
C1 — C2×C4 — C2×C4.10C42
Jennings
C1 — C2 — C2 — C22×C4 — C2×C4.10C42

Generators and relations for C2×C4.10C42
 G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 276 in 186 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4.10C42, C22×M4(2), C2×C4.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.10C42, C2×C2.C42, C2×C4.10C42

Smallest permutation representation of C2×C4.10C42
►On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)

(1 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

(1 29 7 31 5 25 3 27)(2 24 4 22 6 20 8 18)(9 17 15 19 13 21 11 23)(10 28 12 26 14 32 16 30)
 
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,7,31,5,25,3,27)(2,24,4,22,6,20,8,18)(9,17,15,19,13,21,11,23)(10,28,12,26,14,32,16,30)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,29,7,31,5,25,3,27)(2,24,4,22,6,20,8,18)(9,17,15,19,13,21,11,23)(10,28,12,26,14,32,16,30) );
 
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,29,7,31,5,25,3,27),(2,24,4,22,6,20,8,18),(9,17,15,19,13,21,11,23),(10,28,12,26,14,32,16,30)]])
 

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8X
order12222···244444···48···8
size11112···211112···24···4

44 irreducible representations

dim111112224
type++++--
imageC1C2C2C4C4D4Q8Q8C4.10C42
kernelC2×C4.10C42C4.10C42C22×M4(2)C22×C8C2×M4(2)C22×C4C22×C4C24C2
# reps14312126114

Matrix representation of C2×C4.10C42 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
0013000
0001300
0000130
0000013
,
10160000
1670000
000010
000001
0013000
000400
,
0130000
400000
000100
004000
0000013
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,13,0] >;
 

C2×C4.10C42 in GAP, Magma, Sage, TeX 

C_2\times C_4._{10}C_4^2
 
% in TeX
 
G:=Group("C2xC4.10C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,463);
 
// by ID
 
G=gap.SmallGroup(128,463);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,248,1411,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

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